This article is a discovery-based instructional resource for faculty to use as a capstone course or exploratory project for undergraduates who are familiar with (but not necessarily fluent in) calculus, linear algebra, complex variables, and geometry. The explorations flow from a simple change in sign: in the complex numbers a + bi, where…

Every object we think of or encounter, whether a natural or human-made, has a regular or irregular shape. In its own intrinsic conceptual design, it has elements of mathematics, science, engineering, and arts, etc., which are part of the object's geometric shape, form and structure. Geometry is not only an important part of mathematics, but it is…

Questions which stimulate discovery learning are asked concerning polygons and curves drawn on an integral lattice. (DT)

This activity involved constructing parallelograms on the sides of an arbitrary triangle and discovering a relationship between the areas of the parallelograms. (MP)

The process of investigating some of the properties of diagonals and other geometric concepts with a group of 11-year-old girls is presented. Possible changes that could have been made in the lesson are discussed, and examples of individual student work are provided. (MP)

The validity of the formula for the sum of the measures of the interior angles of polygon is explored using non-convex polygons. (SD)

This study is one of several conducted by the authors in their investigation of the use of "higher order rules" in the solution of problems. The focus of the current experiment was determination of the compatibility of identified rules with the knowledge of average teenagers, and of the extent to which instruction in higher order rules…

Having students find and cut out of graph paper the 12 pentominoes can provide an enjoyable exercise in perception, discovery, and double checking. The pentominoes can then be used to illustrate informally several geometric concepts. (SD)

A number of paper-folding activities are described. The activities illustrate geometric concepts, theorems, and construction techniques. (SD)

Examples of activities involving geometric construction are suggested for discovery-oriented instruction. (SD)

Presents an extension of Pick's theorem for simple closed polygonal regions to unions of simple closed polygonal regions. Students are guided to discover Pick's theorem from sets of data including numbers of boundary points and numbers of interior points. (Author/MKR)

An activity-based unit on congruence and similarity is described. The themes underlying the unit are the usefulness of analogy in determining mathematical principles, and the notion that the number of conditions imposed on figures affects the number of factors which must be specified to insure congruence or similarity. (SD)

The number of triangles having a fixed length of longest side is investigated. (SD)

Two students (aged 10) and a teacher described their exploratory approach to problems involved in sliding blocks games. (SD)

Geometric concepts and theorems can be discovered by high school students using the materials described. Among the topics explored are Platonic solids and Euler's formula. (SD)